Explore the concepts of:
Up until now, our models only included fixed effects.
\[ y_{ij} = \mu + \alpha_{i} + e_{ij} \]
\[ y_{ij} = \mu + \alpha_{i} + e_{ij} \]
\[ \alpha_{i} \sim iidN(0, \sigma^2_{\alpha}) \]
\[ e_{ijk} \sim iidN(0, \sigma^2_{e}) \]
The variance of an observation is expressed as:
\[ \sigma^2_{y} = \sigma^2_{a} + \sigma^2_{e} \]
\[\sigma^2_{a}\] group variance: attributed to variability between N rates
\[\sigma^2_{e}\] residual variance: attributed to variability within N rates
Are you interested in specific levels of a factor? Fixed
Are you interested in using levels as a sample of levels from the population, with the goal of assessing variability (and not mean effect) at the population level? AND
Were your levels randomly selected from a population of potential levels? AND
You have sufficient number of levels (>5-8)? Random
Reliably estimating variance components require more data than reliably estimating means
If has < 5-8 levels, then variance estimates would not be accurate, may be best to treat as fixed.
Classic example: blocks in an RCBD.
\[ y_{ij} = \mu + \rho_{j} + \alpha_{i} + e_{ijk} \]
\[ \alpha_{i} \sim iidN(0, \sigma^2_{\alpha}) \]
\[ e_{ijk} \sim iidN(0, \sigma^2_{e}) \]
Similarly to random-effect models, the variance of an observation is expressed as:
\[ \sigma^2_{y} = \sigma^2_{a} + \sigma^2_{e} \]
In our previous RCBD exercise, we analyzed RCBD with blocks as fixed effects, which made our model be a fixed-effect ANOVA model.
Now, let’s treat blocks as random and have a mixed-effect ANOVA model instead (N and K rates are still treated as fixed).